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On partition functions of SCFTs of class $\mathcal{S}$ (Gaiotto theories)

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For which theories of class $\mathcal{S}$ can we write down partition functions (as we can for Lagrangian theories), either in 4d or in the dual 2d CFT description? 

What is known about the $SL(2,\mathbb{Z})$ properties of the partition functions and what is the current progress on this direction?

Any specific references are welcome.

asked Sep 19 in Theoretical Physics by conformal_gk (3,575 points) [ no revision ]

Let me comment that there has been a lot of work for the supersymmetric indices of such theories on $\mathbb{R}^3 \times \mathbb{S}^1$ and $ \mathbb{S}^3  \times \mathbb{S}^1 $. What is the difficulty though for the 4d partition functions?

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